$ -3.\overline{93} \div 0.\overline{7} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 100x &= -393.9394...\\ x &= -3.9394...\end{align*} $ $\begin{align*} 99x &= -390 \\ x &= -\dfrac{390}{99}\end{align*} $ $\begin{align*} 10y &= 7.7777...\\ y &= 0.7777...\end{align*} $ $\begin{align*} 9y &= 7 \\ y &= \dfrac{7}{9}\end{align*} $ So, the problem becomes: $ -\dfrac{390}{99} \div \dfrac{7}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{390}{99} \times \dfrac{9}{7} = {?} $ $ \phantom{-\dfrac{390}{99} \times \dfrac{7}{9}} = \dfrac{-390 \times 9}{99 \times 7} $ $ \phantom{-\dfrac{390}{99} \times \dfrac{7}{9}} = \dfrac{-390 \times \cancel{9}} {\cancel{99}11 \times 7} $ $ \phantom{-\dfrac{390}{99} \times \dfrac{7}{9}} = -\dfrac{390}{77} $